Perturbation methods and first-order partial differential equations on Riemannian manifolds

In this paper, we give explicit estimates that ensure the existence of solutions for first-order partial differential operators on compact manifolds, using a viscosity method. In the linear case, an explicit integral formula can be found, using the characteristic curves. The solution is given explicitly on the critical points and the limit cycles of the vector field of the first-order term of the operator. In the nonlinear case, a generalization of the Weitzenböck formula provides pointwise estimates that ensure the existence of a solution, but the uniqueness question is left open. Nevertheless we prove that uniqueness is stable under a C
¹
perturbation. Finally, we give some examples where the solution fails to exist globally, justifying the need to impose conditions that guarantee global existence. The final result reveals that the zero-order term in the first-order operator is necessary to obtain generically bounded solutions.